We compare Mishchenko's definition of non-commutative signature for an oriented manifold with an orientation preserving proper action of a discrete, countable group $G$ with the (more analytical) counterpart defined by Higson and Roe in the series of articles “Mapping surgery to analysis”. A generalization of the bordism invariance of the coarse index is also addressed.

We investigate a fractional chemotaxis system coupled with the Navier-Stokes fluid in the whole space $\mathbb{R}^{N}$ with $N\geq 3$. With the help an appropriate functional space, we develop a framework for a unified treatment of the existence, uniqueness and decay estimates of global mild solutions to this problem under the assumption that initial data are small enough.

Equivalents of the Brouwer fixed point theorem are proved. They involve formulations either for the standard simplex or for the cube. Characterizations of continuous functions defined on the standard simplex are also presented. The famous Steinhaus chessboard theorem is generalized.

In this paper, with the aid of $L_{0}$-index iteration theory, the minimal period estimates are considered on brake orbits of nonlinear $N$-symmetric Hamiltonian systems with a mild superquadratic growth condition.

In this paper we study to what extent properties of the classical linear Volterra operators can be transferred to the nonlinear Volterra-Choquet operators, obtained by replacing the classical linear integral with respect to the Lebesgue measure, by the nonlinear Choquet integral with respect to a nonadditive set function. Compactness, Lipschitz and cyclicity properties are studied.

We study the multiplicity of semiclassical states for the Choquard equation $$ -\varepsilon^2\Delta u +V(x)u =\varepsilon^{\mu-N}\bigg(\int_{\mathbb{R}^{N}} \frac{G(y,u(y))}{|x-y|^\mu}dy\bigg)g(x,u) \quad \mbox{in $\mathbb{R}^{N}$}, $$ where $0< \mu< N$, $N\geq3$, $\varepsilon$ is a positive parameter and $G$ is the primitive of $g$ which is of critical growth due to the Hardy-Littlewood-Sobolev inequality. The potential function $V$ is assumed to be nonnegative with $V(x)=0$ in some region of $\mathbb{R}^{N}$. Using the genus theory we prove the multiplicity of semiclassical states for the critical Choquard equation.

In this paper, we study a class of singularly perturbed critical quasilinear Schrödinger equation of the form $$ -\varepsilon^2\Delta u+V(x)u-\varepsilon^2(\Delta u^2)u=P(x)|u|^{p-2}u+Q(x)|u|^{2\cdot2^*-2}u,\quad \hbox{in } \mathbb{R}^N. $$ By using a change of variables and variational argument, we prove not only the existence of positive ground state solutions and their concentration behavior, but also the existence and associated concentration behavior of multiple solutions.

In this paper we analyze the asymptotic behavior of the pullback attractors for non-autonomous dynamical systems generated by a family of non-autonomous damped wave equations when some reaction terms are concentrated in a neighbourhood of the boundary and this neighbourhood shrinks to boundary as a parameter $\varepsilon$ goes to zero. We show the gradient-like structure of the limit pullback attractor, the existence and continuity of global hyperbolic solutions and the lower semicontinuity of the pullback attractors at $\varepsilon=0$. Finally, we obtain the continuity of the pullback attractors at $\varepsilon=0$.

In this paper, we consider the existence of ground-state solutions to nonlinear Kirchhoff-type transmission problems by using the methods from (Silvia Cingolani and Tobias Weth, , Ann. Inst. H. Poincaré Anal. Non Linéaire (2016), no. 1, 169-197). Here, we avoid the conditions under which the nonlinear terms of $f$ and $g$ are forms of $C^1$. In particular, when $N=2$, the existence of ground-state solutions is established to the Kirchhoff-type transmission problem with exponent-type nonlinearity.

We analyse positive solutions to the steady state reaction diffusion equation: \begin{equation*} \begin{cases} -u''=\lambda h(t) f(u) \quad \text{in } (0,1), \\ -du'(0)+\mu(\lambda) u(0)=0,\\ u'(1)+\mu(\lambda) u(1)=0, \end{cases} \end{equation*} where $\lambda>0$ is a parameter, $d\geq 0$ is a constant, $f \in C^2([0,\infty),\mathbb{R}) $ is an increasing function which is sublinear at infinity $\Big (\lim\limits_{s \rightarrow \infty}{f(s)}/{s}=0\Big)$, $h \in C^1((0,1],(0,\infty))$ is a nonincreasing function with $h_1:=h(1)>0$ and there exist constants $d_0>0$, $\alpha \in [0,1)$ such that $h(t)\leq {d_0}/{t^\alpha}$ for all $t \in (0,1]$, and $\mu \in C([0,\infty),[0,\infty))$ is an increasing function such that $\mu(0)\geq 0$. We consider three cases of $f$, namely, $f(0)=0$, $f(0)> 0$ and $f(0) <0$. We will discuss existence and multiplicity results via the method of sub-supersolutions. Further, we will establish uniqueness results for $\lambda\approx 0$ and $\lambda\gg 1$.

We consider the following nonlinear elliptic systems: \begin{equation} \left\{\begin{alignedat}2 -\Delta u_1 +\lambda_1u_1 & = \mu_1u_1^3+\beta_{12}u_1u_2^2+\beta_{13}u_1u_3^2 &\quad& \hbox{in } \Omega,\\ -\Delta u_2 +\lambda_2u_2 & = \mu_2u_2^3+\beta_{12}u_2u_1^2+\beta_{23}u_2u_3^2 &\quad &\hbox{in } \Omega,\\ -\Delta u_3 +\lambda_3u_3 & = \mu_3u_3^3+\beta_{13}u_3u_1^2+\beta_{23}u_3u_2^2 &\quad& \hbox{in } \Omega,\\ \overrightarrow{u} & =(u_1,u_2,u_3)\in H^1_0(\Omega)^3, \end{alignedat}\right.\tag{$*$} \end{equation} where $\Omega\subset \mathbb R^n$ ($n\leq 3$) is a bounded domain with smooth boundary, $\lambda_j, \mu_j> 0$ $(j=1,2,3)$, $\beta_{12}> 0$, and $\beta_{13}, \beta_{23}\leq 0$. For this model case of coupled Schrödinger equations with mixed coupling, for sufficiently large $\beta_{12}> 0$, we show that there exists a solution $(u_{1\beta},u_{2\beta},u_{3\beta})$ of $(*)$ such that $u_{1\beta}> 0, u_{2\beta}> 0$ and $u_{3\beta}$ changes sign exactly once. We also show that, for any given $k\in \mathbb{N}$, there exist $k$ vector solutions $(u_{1\beta}^\ell,u_{2\beta}^\ell,u_{3\beta}^\ell)$ $(\ell=1,\ldots,k)$ and these solutions are characterized by the genus with respect to a partial symmetry $\sigma(u_1,u_2,u_3)=(-u_1,-u_2,u_3)$.

In this paper, we consider the following fractional Kirchhoff-Schrödinger equation: \begin{equation*} \begin{cases} \displaystyle \Big(a+b\|u\|_{E}^{p(\theta-1)}\Big) \big[(-\triangle)_{p}^{s}u+V(x)|u|^{p-2}u\big]=f(x, u), & x \in \mathbb{R}^{N},\\ \displaystyle \|u\|_{E}^p:=\iint_{\mathbb{R}^{2N}} \frac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}dxdy +\int_{\mathbb{R}^{N}}V(x)|u|^{p}dx, \end{cases} \end{equation*} where $a> 0$, $b \geq 0$, $\theta \geq1$, dimension $N=sp$ with $s \in (0, 1)$ and $p\geq 1$. $ V$ is a positive potential and $f$ is a critical nonlinearity with exponential growth. We derive a positive ground state solution by using minimax techniques combined with the fractional Truding-Moser inequality. Moreover, in the particular case of $a=1$ and $b=0$, we also obtain the existence of the ground state solution to the fractional Schrödinger equation.

In this paper, we study the multiplicity of periodic solutions of a Duffing equation $$ x''+g(x)=p(t). $$ By using the generalized Poincaré-Birkhoff fixed point theroem, we prove that this equation has infinitely many periodic solutions provided $g$ satisfies a kind of integral condition and the related time map satisfies oscillating condition.

In this paper, we study the following class of Schrödinger equations: \[ -\Delta_{N}u+V(|x|)|u|^{N-2}u=Q(|x|)h(u) \quad \text{in } \mathbb{R}^N, \] where $N\geq 2$, $V,Q\colon \mathbb{R}^{N}\rightarrow \mathbb{R}$ are potentials that can be unbounded, decaying or vanishing at infinity and the nonlinearity $h\colon \mathbb{R}\rightarrow \mathbb{R}$ has a critical exponential growth concerning the Trudinger-Moser inequality. By using a variational approach, a version of the Trudinger-Moser inequality and a symmetric criticality type result, we obtain the existence of nonnegative weak and ground state solutions for this class of problems and under suitable assumptions, we obtain a nonexistence result.

We study extension theorems for Lipschitz-type operators acting on metric spaces and with values on spaces of integrable functions. Pointwise domination is not a natural feature of such spaces, and so almost everywhere inequalities and other measure-theoretic notions are introduced. We analyze Lipschitz-type inequalities in two fundamental cases. The first concerns almost everywhere pointwise inequalities, while the second considers dominations involving integrals. These Lipschitz-type inequalities provide a suitable frame to work with operators that take values on Banach function spaces. In the last part of the paper we use some interpolation procedures to extend our study to interpolated Banach function spaces.

In this paper, we study the asymptotic behavior of the following extensible beam equations: $$ \varepsilon(t) u_{tt}+\Delta^2 u-M\bigg(\int_\Omega |\nabla u|^2\,dx\bigg) \Delta u +\alpha u_t+\varphi (u)=f, \quad t>\tau, $$ where $\varepsilon(t)$ is a decreasing function of time vanishing at infinity. After generalizing the abstract results on time dependent space, we establish an invariant time-dependent global attractor for the equation by proving the well-posedness (thereby, the existence of process), dissapativity and the compactness of the process. Our work supplements the theoretical results on time-dependent space and the results on the longtime behavior of the model.